10 And A <sub> 1 </ Sub> = 1 Distance-regular Graphs,

In this thesis, we give a classification of distance-regular graph of k = 10 and a₁ = 1. By mean of intersection diagrams, circuit chasing techniques, properties of distance-regular graph and the known conclusions of distance-regular graph, we prove the following conclusions.Let Γ be a distance-regular graph of k = 10, a₁ = 1 and r = r(Γ).If c_r+1 = 4, then a(r+1) = 4, d = r + s + 1 and c_d = 5, in this case Γ is a 1-homogeneous graph.If c_r+1 = 3, then T doesn't exist.If c_r+1 = 2 and a_r+1 = 2 , then one of the following holds:1. c_r+2 = 5,d = r + 2;2. c_r+2 = 4 , and one of the following hold:(a) a_r+2 = 5, d — r + t₁ + 4 and c_d = 10;(b) a_r+2 = 4, d = r + t + 3 and c_d = 5,in this case Γ is a 1-homogeneous graph;3. c_r+2 = 3, and one of the following hold:(a) a_r+2 = 7 and d = r + 3;(b) a_r+2 = 5;(c) a_r+2 = 4,d = r + 3 and c_r+3 = 6 or 8;(d) a_r+2 = 3,d = r+t₂ + 5 and c_d = 10 or d = r +t₂ + 4 and c_d = 5 , in this case Γ is a 1—homogeneous graph.